Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms

Abstract

We study mixed surfaces, the minimal resolution S of the singularities of a quotient (C × C)/G of the square of a curve by a finite group of automorphisms that contains elements not preserving the factors. We study them through the further quotientsAs first application we prove that if the irregularity is at least 3, then S is also minimal. The result is sharp.The main result is a complete description of the Albanese morphism of S through a determined further quotient (C ×C)/G ′ that is anétale cover of the symmetric square of a curve. In particular, if the irregularity of S is at least 2, then S has maximal Albanese dimension.We apply our result to all the semi-isogenous mixed surfaces of maximal Albanese dimension constructed by Cancian and Frapporti, relating them with the other constructions of surfaces of general type having the same invariants in the literature.

Topological types of actions on curves

“…The converse is obvious. This fact is very important in the applications to the construction of new deformation types of algebraic varieties as in [1,2,34,33,38,10,30,14,19,3,37,36]. It follows from this discussion that the classification of universal families and of deformation equivalence classes of G-curves are both equivalent to the classification of topological types.…”

A note on a family of surfaces with $$p_g=q=2$$ and $$K^2=7$$

“…In Table 1 have indicated, where possible, the number of families (#) and the dimensions of the irreducible component containing them (dim). Moreover, we point out if some members of the family are product-quotient surfaces (pq) or mixed surfaces (ms), in particular for some PP2 surfaces the proof is given by the second author in [Pig20], while for the remaining ones is unknown. In the last column, we give references to more detailed descriptions of the class.…”

“…Surfaces with p g = q = 4 and p g = q = 3 are completely classified, see [3,11,20,33]. On the other hand, for the the case p g = q = 2, which presents a very rich and subtle geometry, we have so far only a partial understanding of the situation; we refer the reader to [12][13][14][15]25,[27][28][29][30][31][32]34,38] for an account on this topic and recent results.…”

On the minimal model of semi-isogenous mixed surfaces

“…In this section we recall the definition of a semi-isogenous mixed surface and a few general results on them; we refer to [CF18,Pig20] for further details.…”